# MAT 521 Geometry for Teachers

This is a sample syllabus only. Ask your instructor for the official syllabus for your course.

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### Revised Course Description

Topics from geometry including: points and lines in a triangle, properties of circles, collinearity, concurrence, transformations, arithmetic and geometric means, isoperimetric theorems, and reflections principle. Moreover, this course is the study of geometry as an axiomatic system, which includes the study of postulates, theorems, formal proofs, rules of congruence, angle measurement, similarity, parallelism, and perpendicularity. Furthermore, topics in the research of how students learn geometry and effective teaching methods of geometry will be studied.

MAT 521 meets for three hours of lecture per week.

### Prerequisites

Graduate standing and one year of full time secondary teaching.

### Objectives

After completing MAT 521 the student will

• appreciate the pervasive use and power of reasoning as a part of mathematics
• make and test conjectures
• judge the validity of arguments
• construct simple valid arguments
• construct proofs for mathematical assertions, including indirect proofs
• represent problem situations with geometric models and apply properties of figures
• develop an understanding of geometry as an axiomatic system
• translate between synthetic and coordinate representations
• deduce properties of figures using transformations and using coordinates
• identify congruent and similar figures using transformations
• analyze properties of Euclidean transformations and relate translations to vectors
• be knowledgeable of current technologies relevant to geometry instruction
• be knowledgeable of the current research and theories of geometry instruction.

### Expected outcomes

Students should be able to demonstrate through written assignments, tests, and/or oral presentations, that they have achieved the objectives of MAT 521.

### Method of Evaluating Outcomes

Evaluations are based on problem solving and reasoning performance tasks, homework, projects, papers, class presentations, short tests, portfolio of total work for the semester, and/or scheduled examinations.

1. Brandell, J. (1994). Helping students write paragraph proofs in geometry. Mathematics Teacher, 87(7), 498-502.
2. Burger, W. & Culpepper, B. (1993). Restructuring geometry. In P.S. Wilson (Ed.) Research ideas for the classroom: High school mathematics. (pp. 140-154). New York: MacMillan.
3. Chazan, D., & Houde, R. (1989). How to use conjecturing and microcomputers to teach geometry. Reston, VA: National Council of Teachers of Mathematics.
4. Craine, T. (1985). Integrating geometry into the secondary mathematics curriculum. In C. Hirsch & M. Zweng (Eds.) The Secondary School Mathematics Curriculum (pp. 119-133). Reston, VA: National Council of Teachers of Mathematics.
5. Crowley, M. (1987). The van Hiele model of the development of geometric thought. In M. Lindquist & A. Shulte (Eds.) Learning and Teaching Geometry, K-12 (pp. 1-16). Reston, VA: National Council of Teachers of Mathematics.
6. De Villiers, M. (1998). An alternative approach to proof in dynamic geometry. In R. Leher & D. Chazan (Eds.), Designing learning environments for developing understanding of geometry and space. (pp. 369-393) Mahwah, NJ: L. Erlbuam.
7. Dennis, D. & Confrey, J. (1998). Geometric Curve-drawing devices as an alternative approach to analytic geometry. In R. Leher & D. Chazan (Eds.), Designing learning environments for developing understanding of geometry and space. (pp. 297-318) Mahwah, NJ: L. Erlbuam.
8. Dreyfus, N. (1987). Euclid may stay and even be taught. In M. Lindquist & A. Shulte (Eds.) Learning and Teaching Geometry, K-12 (pp. 47-58). Reston, VA: National Council of Teachers of Mathematics.
9. Geddes, D. & Fortunato, I. (1993). Geometry: Research and classroom activities. In D. T. Owens (Ed.) Research ideas for the classroom: Middle grades mathematics. (pp. 199-224). New York: MacMillan.
10. Goldenberg, E.P. & Cuoco, A. (1998). What is dynamic geometry? In R. Leher & D. Chazan (Eds.), Designing learning environments for developing understanding of geometry and space. (pp.351-367) Mahwah, NJ: L. Erlbuam.
11. Gravemeijer, K. (1998). From a different perspective: Building on students' informal knowledge. In R. Leher & D. Chazan (Eds.), Designing learning environments for developing understanding of geometry and space. (pp.45-66) Mahwah, NJ: L. Erlbuam.
12. Koedinger, K. (1998). Conjecturing and argumentation in high-school geometry students. In R. Leher & D. Chazan (Eds.), Designing learning environments for developing understanding of geometry and space. (pp. 319-347) Mahwah, NJ: L. Erlbuam.
13. Lampert, M. (1993). Teachers' thinking about students' thinking about geometry: The effects of new teaching tools. In J. Schwartz, M. Yerushalmy, & B. Wilson. (Eds.), The geometric supposer: What is it a case of? (pp. 143-178) Hillsdale, NJ: Lawrence Erlbaum Associates.
14. MacPherson, E. (1985). The themes of geometry: Design of the nonformal geometry curriculum. In C. Hirsch & M. Zweng (Eds.) The Secondary School Mathematics Curriculum (pp. 65-80). Reston, VA: National Council of Teachers of Mathematics.
15. Niven, I. (1987). Can geometry survive in the secondary curriculum? In M. Lindquist & A. Shulte (Eds.) Learning and Teaching Geometry, K-12 (pp. 37-46). Reston, VA: National Council of Teachers of Mathematics.
16. Senk, S. (1989). Van Hiele levels and achievement in writing geometry proofs. Journal for Research in Mathematics Education, 20(3), 309-321.
17. Steen, L.A. (1999). Twenty questions about mathematical thinking. In L.V. Stiff & F. R. Curcio (Eds.), Developing mathematical reasoning in grades K-12. (pp. 270-286). Reston, VA: NCTM.
18. Toumasis, C., (1994). When is a quadrilateral a parallelogram? Mathematics Teacher, 87(3), 208-211.
19. Usiskin, A. (1987). Resolving the continuing dilemmas in school geometry. In M. Lindquist & A. Shulte (Eds.) Learning and Teaching Geometry, K-12 (pp. 17-31). Reston, VA: National Council of Teachers of Mathematics.
20. Yerushalmy, M. (1993). Generalization in geometry. In J. Schwartz, M. Yerushalmy, & B. Wilson (Eds.), The geometric supposer: What is it a case of? (pp. 57-84). Hillsdale, NJ: Erlbaum.
21. A collection of assignments handed out in class.

#### Course Content

• Generalization in geometry
• The van Hiele model of the development of geometric thought
• Methods of Proof
• Mathematical reasoning
• Axiomatic systems
• Congruent triangles
• Perpendicular and Parallel lines Quadrilaterals
• Problem posing and conjectures
• Area
• Similarity
• Circles
• Regular Polygons and Circles
• Concurrence Theorems
• Coordinate Geometry
• Dynamic geometry
• Real world geometric applications

Students' grades may be based on homework, projects, papers, class presentations, short tests, and/or scheduled examinations that test students' understanding of the topics covered in the course (see "Method of evaluating outcomes"). The instructor determines the weight of each of these factors in the final grade.

### Attendance Requirements

Attendance policy is set by the instructor.

### Policy on Due Dates and Make-Up Work

Due dates and policy regarding make-up work are set by the instructor.

### Schedule of Examinations

The instructor sets all test dates except the date of the final exam. The final exam is given at the date and time announced in the Schedule of Classes.